In this introductory chapter, we present a number of elementary concepts
and propositions on semigroups, most of which will be indispensable for the
remainder of the book. This chapter has also been written with the aim of
giving the casual reader a broad survey of the subject which is also well-
rounded and not too superficial. This explains why certain topics are treated
here, which could well have been postponed to later chapters (especially in the
later sections of this chapter).
A binary operation on a set S is a mapping oiSxS into S, where S x 8 is the
set of all ordered pairs of elements of 8. If the mapping is denoted by a dot
(•), the image in S of the element (a, b) of S x S (a and b in S) will be denoted
by a-b. Frequently we shall omit the dot, writing ah for a b. Other
symbols which we may use to denote binary operations are -f, °, and *.
A groupoid is a system 8( •) consisting of a non-empty set 8 together with a
binary operation (•) on S. We shall usually write 8 instead of S( •) when
there is no danger of ambiguity.
A partial binary operation on a set 8 is a mapping of a non-empty subset of
8 x S into S. By a partial groupoid we shall mean a system S( •) consisting
of a non-empty set 8 together with a partial binary operation (•) on S.
A binary operation (•) on a set S is called associative if a-(b-c) = (a-b)-c
for all a, 6, c in S. A semigroup is a groupoid S( •) such that the operation
(•) is associative. We frequently use the expression, "S is a semigroup with
respect to (•)," to mean that (•) is an associative binary operation on 8.
Frequently this is further abbreviated to "S is a semigroup".
The object of investigation of this book is semigroups and not groupoids
or partial groupoids. The latter more general systems are, however, occa-
sionally useful in the theory of semigroups, and so must be taken into account.
One exception we shall make to the above terminology is the concept of
Brandt groupoid (§3.3), which is really a partial groupoid satisfying several
rather stringent conditions.
By a transformation of a set X we shall mean a single-valued mapping of X
into itself. Except in Chapter 5, we shall denote the image of an element
x of X under a transformation or mapping a by xa (rather than ax or a(x)).
By the product (or iterate or composition) of two transformations a and £ of X
we mean the transformation aj8 defined by x(afi) = (xa)j3 for all x in X (that
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